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Complete characterization of planar C1 linearizability and optimal conjugacy regularity

  • Zhicheng Tong
  • Yong Li
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Abstract

We establish a complete characterization of local C1 linearizability for planar diffeomorphisms. For every prescribed hyperbolic real Jordan form, we obtain necessary and sufficient integrability conditions on the modulus of continuity of the derivative for local C1 linearizability. These criteria reduce to three canonical thresholds: the classical, polynomially weighted, and squared-logarithmically weighted Dini conditions. Each threshold is optimal: whenever the corresponding condition fails, we construct a diffeomorphism with the identical linear part that is not locally C1 linearizable. For every non-hyperbolic linear part, explicit polynomial counterexamples show that smoothness alone cannot guarantee even local topological linearization. Beyond C1 existence, we determine the optimal regularity for the derivative of the linearizing conjugacy in every hyperbolic case, with the optimality established by counterexamples.

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Cite

BibTeXRIS
BibTeX
@misc{tong2026complete,
  title = {{Complete characterization of planar $C^1$ linearizability and optimal conjugacy regularity}},
  author = {Zhicheng Tong and Yong Li},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21233},
  eprint = {2609.21233},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21233},
}
RIS
TY  - UNPB
TI  - Complete characterization of planar C^1 linearizability and optimal conjugacy regularity
AU  - Zhicheng Tong
AU  - Yong Li
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21233
UR  - https://arxiv.org/abs/2609.21233
AB  - We establish a complete characterization of local C^1 linearizability for planar diffeomorphisms. For every prescribed hyperbolic real Jordan form, we obtain necessary and sufficient integrability conditions on the modulus of continuity of the derivative for local C^1 linearizability. These criteria reduce to three canonical thresholds: the classical, polynomially weighted, and squared-logarithmically weighted Dini conditions. Each threshold is optimal: whenever the corresponding condition fails, we construct a diffeomorphism with the identical linear part that is not locally C^1 linearizable. For every non-hyperbolic linear part, explicit polynomial counterexamples show that smoothness alone cannot guarantee even local topological linearization. Beyond C^1 existence, we determine the optimal regularity for the derivative of the linearizing conjugacy in every hyperbolic case, with the optimality established by counterexamples.
ER  -